NASA Technical Reports Server (NTRS) 19960008823: The minimal time detection algorithm

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(NASA-CR-199324) THE MINIMAL TIME N96-15989
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1
The Minimal Time Detection Algorithm ^ /
Sungwan Kim
Electrical Engineering Department
University of California, Los Angeles
Abstract — An aerospace vehicle may operate throughout a wide range of flight environ-
mental conditions that affect its dynamic characteristics. Even when the control design
incorporates a degree of robustness, system parameters may drift enough to cause its
performance to degrade below an acceptable level. The object of this paper is to develop
a change detection algorithm so that we can build a highly adaptive control system appli-
cable to aircraft systems. The idea is to detect system changes with minimal time delay.
The algorithm developed is called Minimal Time-Change Detection Algorithm(MT-CDA)
which detects the instant of change as quickly as possible with false-alarm probability be-
low a certain specified level. Simulation results for the aircraft lateral motion with a
known or unknown change in control gain matrices, in the presence of doublet input, in-
dicate that the algorithm works fairly well as theory indicates though there is a difficulty
in deciding the exact amount of change in some situations. One of MT-CDA distinguish-
ing properties is that detection delay of MT-CDA is superior to that of Whiteness Test.
1. Introduction
We wish to design an aerospace vehicle control system which operates throughout a wide
range of flight environmental conditions that affect its dynamic characteristics. The prob-
lem of identifying aircraft system parameters from flight test data and their application to
flight control has been successfully solved for linear models, by Balakrishnan [2], Iliff [11],
and Maine [15], [16], when there are no severe changes in aircraft and environment. Even
when the control design incorporates a degree of robustness, system parameters may drift
enough to degrade its performance below an acceptable level. In other words, we still
need to develop a theory for a better control system when aircraft flies at a high angle of
attack where the linear model is no longer valid and when the robust controller does not
work due to large variations in parameters or noise levels.
Adaptive controls are considered to be a promising approach to give us a possible
solution. An adaptive control problem arises whenever system parameters are unknown
or are subject to unknown variation which is usually a small (or smooth) change. It is
also of increasing practical importance, since adaptive techniques are being used more
and more in industrial control systems. However, this field is not mature yet, and only a
set of scattered simulational results exists with no single quantitative measure for judging
performances [1]. Taking into account abrupt (or drastic) changes in statistical models
appears as a natural complement of most of the adaptive techniques which track only slow
variations of parameters. So, the detection of abrupt change is essential in the design of
lrThis work was supported by NASA Ames Research Center, DFRF through UCLA Flight Systems
Research Center(FSRC), Grant Number NCC 2-374.
an adaptive system for these types of adaptive control problems. Therefore, the problem
of detection, estimation, and diagnosis of changes in dynamical properties of signals or
systems are considered, with particular emphasis on statistical methods for detection, to
provide a general framework for change detection in signals and systems.
The development of the change detection problem was stimulated by Wald [25] in 1947
when 'Sequential Analysis' was published and a Sequential Probability Ratio Test(SPRT)
was introduced. In 1954, a 'sequential process inspection scheme' was proposed by Page
[18], [19] to detect a change in the mean by testing a weighted sum of the last few obser-
vations, i.e. a moving average. Named the 'Page-Hinkley' rule, Page pointed out that this
rule is equivalent to performing a SPRT. The theoretical properties of this rule have been
investigated for a long time from on-line and off-line points of view. The most significant
works in that direction are Shiryayev [21] and Lorden [14]. They formulated an 'Opti-
mal Stopping (or Detection)' problem and proposed simple rules which are optimal (or
asymptotically optimal) in an appropriate sense. And the general problem of optimality
of random processes in discrete time was considered and developed by Chow, Robbins,
and Siegmund [7]. The problem of change detection has received growing attention during
the last 20 years, as can be seen from the survey papers (Wilsky [27] and Basseville [4]),
the books (Basseville and Benvensite [5], Basseville and Nikiforov [6]) and so on. Further
discussions may be found in these References.
The development of change detection method is still a relatively new subject. Presently,
most of the work has been at a theoretical level with only a few real applications of tech-
niques: edge detection in pattern recognition, segmentation of speech (or image) signal,
biomedical signal monitoring, vibration monitoring, quality control, fault-detection, etc.
As mentioned earlier, an algorithm is required to adapt to new circumstances for a
total adaptive system applicable to the aircraft control systems. One important new
idea is an algorithm to detect system change with minimal time delay that involves fairly
advanced stochastic theory. This could be formulated based on the problem of an optimal
stopping rule [21] in which we seek to detect the instant of change as quickly as possible,
subject to the given false-alarm probability.
A block diagram shown in Figure 1 explains the total adaptive system for the aircraft
control that is equipped with a CDA. Before the CDA detects a change, the regulator is
designed based on the initial parameter values. But after the change is detected at time
T, the CDA block generates an Activation Signal to re-initiate system identification. The
system identification process finds a new set of parameter values proper for the changed
system. The regulator uses the new parameter values to build a new control law. This
process is continued until the new parameter set converges. So, a total adaptive control
system can be designed that is good for a system that undergoes changes.
Algorithm complexity is definitely an issue for detection of each possible change and
time of change. In spite of this complexity, we can have a control system which can detect
its change and adapt it as quickly as possible even though the large amount of change is
occurred at random time. Therefore, this MT-CDA will present a proper starting point
for total adaptive control system in the theoretical point of view.
Nolle
Commend
Measured Output
Figure 1: A Block Diagram of Control System for Aircrafts with CDA
2. Minimal Time-Change Detection Algorithm
2.1. MT-CDA: Detection of a Known Change
The following state space model is considered.
= Ax + (B + (B - £i)x(n - 6}}u
n l 2 n (1)
vn = Cxn (D-2 - GNn. (2)
-{
0 if m < 0
(3)
1 if m > 0
Based on the Eq.(l) and (2), the basic problem to be solved is to detect a known change in
control gain matrices(£? and D) as quickly as possible in order to build an highly adaptive
system whose parameters(B and D) could be switched from a known system to another
known system after change detection. This is appropriate for a system which operates
in two alternate driving points, and restricted to the class of models with a change of
parameters which is known. The probability of making a false (or wrong) detection also
needs minimized since the signals are disturbed by a white Gaussian noise.
We want to formulate a problem which is to find the detection time(r) to minimize
the average, delay
E[(r - 0)+} (4)
subject to a given false alarm probability, i.e.
Pr\T < 0] < a (5)
where a is given. 'Likelihood Ratio (change vs. no change)', which we shall define to be
the ratio
Joint density of ui, • • •, v (with change)
n
Joint density of Ui, • • • ,u (without change)
n
is introduced and p(-), Poo(-) denote the likelihood probability density function with and
without changes at 0 = k.
. p(vi,v ,---,v \0 = k) = p(vi,---,v ,v +i,---,v \0 = k) (6)
2 n k k n
--,v \0 > n) (7)
n
Then we define the following functional
L k = 2 log ''-j-1 — —. (8)
n<
and for each n,
4 = maxl , . (9)
n fc
Now a decision rule for CDA is
T = min{n : £ > h} (10)
n
where r is a change detection time and h is a threshold value computed from the proba-
bility of false alarm equation(Eq.(5)).
This algorithm stops the first time when £ exceeds a given threshold. This is based
n
on Eq.(8)-(10), so it can be regarded as 'Maximum Likelihood' treatment of the unknown
change time, i.e. stop at r = n when for some k the observations vk, • • •, v are significant.
n
By the contributions of earlier works of Shiryayev [21], Lorden [14], Bansal [3], and
Tartakovsky [23], [24] this procedure could be said optimal (at least asymptotically op-
timal), in the sense that it minimizes the detection delay for a fixed false alarm probability.
2.2. MT-CDA: Detection of an Unknown Change
In this section, minimal time detection of an unknown change instead of a known change
is considered, and its algorithm is derived. The detection of an unknown change is a more
complicated problem than the detection of a known change because there is no information
on the amount of change. Multiple detectors are required to detect an unknown change,
where the each detector is based on the known amount of change, in contrast to single
detector required for the detection algorithm of a known change. In Section 2.2.1, the
algorithm for the detection of only change time, which is a little bit simpler problem, is
first derived and the detection algorithm of both time and amount of change is followed
in Section 2.2.2.
2.2.1. Detection of the Time (or Instant) of Change: The detection of change time is
a significant problem in some applications, for example,
1. Fault detection : all the processes are stopped when a failure occurs and are re-
started after the failure is fixed.
2. When a system requires the exact parameter set after a change, it is preferable to
re-initiate the system identification just after detecting a change.
In adaptive system sense, the 2nd case is a popular one and a promising problem to
be considered [9]. To detect an unknown change, the CD A Block in Figure 1 contains
multiple change detectors. Each of (CDA),- is based on a fixed known change, so each of
detectors is basically the same one as derived in Section 2.1. By examining the Activation
Signal for each n, the real Activation Signal at time n is generated by the earliest one. The
idea of this algorithm is that if there exists an unknown change, the exact (or the closest)
detector will make Activation Signal at the earliest time. So the time of an unknown
change will be detected with this algorithm if the amount of an unknown change is in the
range of the multiple detector hypotheses.
To formulate this MT-CDA for an unknown change, we consider the following system
model
x = Ax + (B + (B - Bi)x(n - 9}}u (11)
n+l n l 2 n
v = Cx + (D + (D - Di)x(n - 0)K + GN (12)
n n l 2 n
#2) DI • • • unknown.
The only difference, compared to the Section 2.1, is that control gain matrices(jB and
2
D?) are unknown which stands for the unknown change. Like Section 2.1, we define the
following functional, which is related to 'Likelihood Ratio', for each detector
U = 2 n
nk l o g :
n'k 6 V '
where each superscript T stands for z'-th detector (i = 1,2, • • • , N) and for each n and
each detector,
. (14)
nn < n.fkc V I
fc n
Then the decision rule for the detection of an unknown change is
r1' = min{n : P > h'} (15)
n
T = minr' (16)
where T* is a change detection time of each detector, r is a real change detection time,
and h* is a threshold value for each detector which is found from the probability of false
alarm equation. So we can detect this kind of unknown change at time r which is the
earliest detection time among all the multiple detectors considered in this algorithm. To
have a good detection algorithm, the number of change detectors(TV) should be sufficiently
large and the dynamic range of detectors' change amount should be sufficient to cover the
estimated range of changes in system parameters. This means that there exists a trade-
off relationship between 'Performance' and 'Complexity' of the algorithm because the
computational burden increases dramatically as the number of change detectors increases.
2.2.2. Detection of the Time and Amount of Change: In some applications, it is
insufficient to detect only the change time if the change is unknown, it is also required to
detect the amount of change. Figure 2 is a modified block diagram of Figure 1 for this
purpose. CDA Block is a change detection algorithm which can estimate the amount of
an unknown change and the change time. It generates the Activation Signal^.?) and
an Estimated Degree of Change(EDC) which is closely related to the real amount of
change limited by the resolution of the detectors. When the system change is detected,
the system parameter set is updated based on EDC and Parameter Update Block that
rules the parameter update law. There is no necessity of System Identification routine.
This looks simpler than the adaptive control system explained in Section 2.2.1, but the
CDA Block is much more complicated than CDA Block. So, Figure 2 could be considered
another version of block diagram for the total adaptive control system that is appropriate
for a system that undergoes an unknown change in parameters. To determine the amount
Turbulence Noise
i
Commend ™ Input Signal
_J Regulator |-| >| Aircraft \ •O1 Measured Output
State Estl.
'
Para. Update 1 | Model
| Computed Output
Aethntlon Signal CDA ' '
EttlmMod Degree of Chinge
Figure 2: A Block Diagram of Control System for Aircrafts with CDA
of change, a 'Decision Mechanism' is required. Based on the AS, each of that comes from
a known change detector, the Decision Mechanism generates a real one and an EDC at
time n. The algorithm formulation is the exactly same as Section 2.2.1 (Eq.(ll)-(16))
except the Decision Mechanism, so only the details of Decision Mechanism is explained
in the following.
T1' = min{n : f > h'} (17)
n
(18)
K = min{z : r'}. (19)
We can then detect an unknown change at time r whose EDC is equivalent to the /c-th
detector, if the amount of an unknown change is in the range of the amount of multiple
detectors. The difficulty of the decision of the exact EDC exists in decision rule(Eq.(17)-
(19)), called 'D.R. # 1' hereafter, because some of change detectors could respond at the
same time. That means the necessity of a better decision rule. Because the exact value
of £l ('exact' means the exact or the closest change detector corresponding to the given
n
unknown change) is rapidly increasing, the following new decision rule, called 'D.R. # 2',
could be proposed.
r' = min{n : f > A1'} (20)
n
r = minrf' (21)
{ max{z : (& — h*)} if 3 more than 2 minimum TI , .
n
min{z : r1} otherwise. ^ '
There exists the same trade-off between performance and the complexity of the algorithm
explained in Section 2.2.1. Only a finite number of change detectors are used in real
simulations.
3. Simulations and Results
A lateral rigid body dynamics of PA-30, which is a Linear Time-Invariant(LTI), is used
for simulations [12], [17].
3.1. Detection of a Known Change
The mean(M[-]) and the variance(V[-]) of detection delay(r — 0], with respect to the
different change time(#), based on 100 runs hereafter, are summarized in Table 1. 50%
change is detected earlier than 25% or 10% change. From the result of Table 1, it is shown
that different known changes could been detected with some delays.
3.2. Detection of an Unknown Change: Time only
Simulations with 6 change detectors are done first and then combined to analyze their
results as summarized in Table 2. From this Table, the followings are observed:
• For a given detector, large amount of change is detected earlier than small amount
of change.
e
10 20 30 40 50 60 70 80 90 100
M[T - 6] 7.6 0.9 7.1 4.0 11.7 10.3 11.4 12.2 11.7 12.1
V(r - 8} 0.4 0.3 4.5 0.4 26.6 8.2 11.7 12.2 13.3 18.2
(a) 10% Change
e
10 20 30 40 50 60 70 80 90 100
M[T - 9} 6.2 0.1 3.1 3.0 1.6 4.6 4.8 5.2 5.1 5.0
V(r - 9] 0.5 0.1 0.6 0.1 1.9 1.0 1.6 1.4 1.8 1.7
(b) 25% Change
0
10 20 30 40 50 60 70 80 90 100
M[r - 9} 4.5 0.0 1.4 2.2 0.2 2.3 2.5 2.8 2.7 2.7
V[T - 0} 0.7 0.0 0.3 0.2 0.1 0.5 0.6 0.5 0.5 0.5
(c) 50% Change
Table 1: Simulation Results for Various Change Times when Change is in Rolling Motion
• Rolling motion :
- 10% Change:50%D > 25%D > 10%D
- 25% Change:10%D,50%D > 25%D =^ 10%D,50%D ~ 25%D
- 50% Change: 10%D > 25%D > 50%D ^ 10%D ~ 25%D ~ 50%D
where 'D' denotes a detector and the detection delay is quite reduced when 10%
detector is used to detect 10% change but there is no big difference in detection
delay of 3 change detectors to detect 25% or 50% change.
• Yawing motion : The results are the same as rolling motion, but detection delay is
large and the difference of delay with respect to each change detector is small since
the portion of yawing motion in PA-30 is small.
From the above observations and the trade-off between the number of change detectors
and the computational burden, it could be concluded that 10% change detector is suffi-
cient when the detection of change larger than 10% is concerned.
3.3. Detection of an Unknown Change: Time and Amount
Simulation results, with D.R. # 2, for 6 = 40 are summarized in Table 3. From this table,
it could be shown that the unknown changes at 0 = 40 are fairly well detected except 30%
1
10% 25% 50%
R.D. Y.D. R.D. Y.D. R.D. Y.D.
10% R.C. 7.05 16.16 9.02 15.84 12.31 15.71
10% Y.C. N.D. 16.61 N.D. 16.61 N.D. 16.49
20% R.C. 3.88 15.61 3.84 15.58 4.31 15.0
20% Y.C. N.D. 16.61 N.D. 16.60 N.D. 16.48
25% R.C. 3.23 15.17 3.11 15.01 3.29 14.91
25% Y.C. N.D. 16.61 N.D. 16.60 N.D. 16.48
30% R.C. 2.73 15.02 2.55 14.88 2.73 15.85
30% Y.C. N.D. 16.61 N.D. 16.59 N.D. 16.47
40% R.C. 2.18 14.79 1.90 14.73 1.94 14.77
40% Y.C. N.D. 16.61 N.D. 16.56 N.D. 16.46
50% R.C. 1.81 14.66 1.57 14.68 1.43 14.70
50% Y.C. N.D. 16.61 N.D. 16.56 N.D. 16.45
60% R.C. 1.55 14.54 1.25 14.63 1.11 14.64
60% Y.C. N.D. 16.56 N.D. 16.51 N.D. 16.45
Table 2: Summarized Simulation Results for 9 = 30
R.D. • • • Rolling Detector
Y.D. • • • Yawing Detector
R.C. • • • Rolling Change
Y.C. • • • Yawing Change
N.D. • • • No Detection
and 40% changes, which might be fixed if the better decision rule is applied. Simulation
results for 6 = 10, 20, and 30 have the same characteristics.
Therefore, it could be concluded that the given unknown changes are detected in sta-
tistical sense, though there exists a difficulty in deciding the exact EDO in some situations
which might be improved by applying the better decision rule.
3.4 Change Detection via Whiteness Test
To examine the validity of MT-CDA, derived in Section 2.1 and 2.2, simulations of White-
ness Test are done and results are compared in this section. Only the detection problem
of a known change is considered here.
Table 4 shows the detection delay of each algorithm for various change time(0). For
every tests, the detection delay of MT-CDA is smaller than that of Whiteness Test.
Therefore, the simulation results show that MT-CDA is superior to Whiteness Test in the
sense of detection delay comparison.